Thermalization and dynamical spectral properties in the quark-meson model
Linda Shen ,1,* Jürgen Berges,1 Jan M. Pawlowski,1 and Alexander Rothkopf2
1Heidelberg University, Institute for Theoretical Physics, Philosophenweg 16, 69120 Heidelberg, Germany
2University of Stavanger, Faculty of Science and Technology, Kristine Bonnevies vei 22, 4036 Stavanger, Norway
(Received 14 March 2020; accepted 8 June 2020; published 16 July 2020)
We investigate the nonequilibrium evolution of the quark-meson model using two-particle irreducible effective action techniques. Our numerical simulations, which include the full dynamics of the order parameter of chiral symmetry, show how the model thermalizes into different regions of its phase diagram. In particular, by studying quark and meson spectral functions, we shed light on the real-time dynamics approaching the crossover transition, revealing, e.g., the emergence of light effective fermionic degrees of freedom in the infrared. At late times in the evolution, the fluctuation-dissipation relation emerges naturally among both meson and quark degrees of freedom, confirming that the simulation approaches thermal equilibrium.
DOI:10.1103/PhysRevD.102.016012
I. INTRODUCTION
The quest to discover the conjectured critical point of the QCD phase diagram is a central motivation of modern heavy-ion collision experiments at collider facilities, such as the Large Hadron Collider at CERN and the Relativistic Heavy-Ion Collider (RHIC) at Brookhaven National Laboratory. In the beam energy scan currently executed at RHIC, the phase diagram of QCD is explored over a wide range of temperatures and baryon densities by depositing different amounts of energy in the initial collision volume.
As the fireball expands and cools, the efficient exchange of energy and momentum among quarks and gluons leads to local thermalization over time. The question to answer is: if a critical point exists and some of the volume of the fireball evolves close to it, does the dynamical buildup of long range fluctuations leave any discernible mark on the yields of measurable particles?
Understanding the out-of-equilibrium dynamics of heavy-ion collisions thus remains one of the most pressing theory challenges in heavy-ion physics. So far, genuinely nonperturbative ab initio calculations of the equilibration process of the quark-gluon plasma and the dynamics close to the phase transition remain out of reach. In order to make
progress, we therefore set out to shed light onto pertinent aspects of the physics of dynamical thermalization in heavy-ion collisions by deploying a low-energy effective theory of QCD, the two-flavor quark-meson model. This model incorporates the off-shell dynamics of the lowest mass states in QCD, the pseudoscalar pions, and the scalar sigma-mode, as well as the light up and down quarks.
Further degrees of freedom, and in particular the gluons, heavier quark flavors, as well as higher mass hadronic resonances carry masses ≳500MeV and are neglected here. This low-energy effective theory reflects the central and physically relevant feature of low-energy QCD: chiral symmetry breaking in vacuum and its restoration at finite temperature and density. At its critical end point, the model is expected to lie in the same universality class as QCD and hence constitutes a viable low-energy effective theory to explore dynamical critical phenomena in QCD at finite temperature and density at scales≲500MeV.
In the present work, we consider the real-time dynamics of the two-flavor quark-meson model with small current quark masses in a nonexpanding scenario; for progress on the out-of-equilibrium quark-meson model, see[1–4]. In the presence of such an explicit chiral symmetry breaking, the equilibrium chiral transition at finite temperature is a crossover as confirmed for QCD at vanishing and small density; for recent results, see [5–7]. By the help of different initial conditions defined via the initial occupa- tions of sigma and quark fields, we map out the thermal- ization dynamics for different regions of the phase diagram.
This allows, for the first time, to study the full thermal- ization dynamics including that of order parameters of
*shen@thphys.uni-heidelberg.de
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tities, they map out the whole time evolution of the system including the physics of the crossover transition in the late-time limit.
This paper is organized as follows. In Sec.II, we briefly review the quark-meson model and give an overview over our nonequilibrium and nonperturbative treatment.
The numerical setup for the time evolution starting from free-field initial conditions quenched to a highly non- equilibrium environment is described. In Sec. III, we discuss the spectral functions of the bosonic and fermionic degrees of freedom, which provide information about the masses as well as the lifetimes of the dynamical degrees of freedom. We investigate the late-time limit of our simu- lations, which reveals the dynamical emergence of the fluctuation-dissipation relation and hence allows us to define a thermalization temperature. Finally, Sec.IVcovers the results for the sigma field describing the order param- eter of the quark-meson model. We further discuss the behavior of different order parameters in equilibrium which lead to a consistent pseudocritical temperature.
In Sec. V, we conclude with a summary. The Appendix provides details about the evolution equations of the model including the relevant expressions for the deployed approximation scheme.
II. THE QUARK-MESON MODEL
QCD evolves from a theory of dynamical quarks and gluons at large momentum scales, the fundamental degrees of freedom, to a theory of dynamical hadrons at low momentum scales. This transition of the dynamical degrees
low-energy effective theories. While its quantitative val- idity has been proven for momentum scales k with k≲ 300MeV [10], it reproduces qualitative QCD features up tok≲700MeV. It is this natural QCD embedding as well as its robust QCD-type chiral properties that has triggered a plethora of works with the quark-meson model on the QCD phase structure with functional methods; see, e.g.,[11–16].
More recently also, real-time correlation functions in equilibrium have been investigated in, e.g.,[17–27].
(Pre-)Thermalization has been studied in theOðN ¼4Þ symmetric scalar model coupled to fermions using a two- loop approximation of the 2PI effective action in[1,28].
The model was studied extensively in Refs. [2,3] in the context of inflaton dynamics to describe nonequilibrium instabilities with fermion production from inflaton decay.
In [4], the model was investigated for highly occupied bosonic fields, where the predictions were shown to agree well with lattice simulation results in the classical-statistical regime. Further results for spectral functions in and out of equilibrium with 2PI effective action techniques can be found in[29], and with classical-statistical simulations in [30–32] for scalar theories, and in [33] for Yang-Mills theory.
In this work, we build on these results and investigate the nonequilibrium evolution of the two-flavor quark-meson model: we consider two light quark flavors with isospin symmetry, up and down quarks with an identical current quark massmu=d¼mψ, coupled to a scalar mesonic fieldσ and a triplet of pseudoscalar pionsπα(α¼1, 2, 3) through a Yukawa couplingg. The classical action reads
S½ψ¯;ψ;σ;π ¼ Z
d4x
¯
ψðiγμ∂μ−mψÞψ− g
Nfψ¯ðσþiγ5ταπαÞψ þ1
2ð∂μσ∂μσþ∂μπα∂μπαÞ
−1
2m2ðσ2þπαπαÞ− λ
4!Nðσ2þπαπαÞ2
; ð1Þ
withτα(α¼1, 2, 3) denoting the Pauli andγμ (μ¼0, 1, 2, 3) the Dirac matrices, while spinor and flavor indices are suppressed, In(1).mψ is the current quark mass, andm2the mesonic mass parameter. The lowest mass states of the
mesonic scalar-pseudoscalar multiplet, σ and ⃗π, are given by the N ¼4 scalar components of the bosonic fieldφaðxÞ ¼ fσðxÞ;π1ðxÞ;π2ðxÞ;π3ðxÞginteracting via a quartic self-couplingλ. The boson fieldsφaare coupled to
the fermion fields ψ and ψ¯ ¼ψ†γ0 via the Yukawa interactiong, which we also express in terms ofh¼g=Nf. The π mesons play the role of the light Goldstone bosons in the chirally broken phase, whereas theσmeson represents the heavy mode. Assigning these roles to the components of the scalar field is achieved by choosing a coordinate system in field space where the field expectation value has a single component which defines theσdirection, i.e.,ϕaðxÞ ¼ hφðxÞi ¼ fhσðxÞi;0;0;0g.
The quasiparticle excitation spectrum of the quark- meson model is encoded in the spectral functions of the respective fields. For the bosonic and fermionic fields, the spectral function is defined as the expectation value of the commutator and anticommutator, respectively,
ρϕabðx; yÞ ¼ih½φaðxÞ;φbðyÞi;
ρψABðx; yÞ ¼ihfψAðxÞ;ψ¯BðyÞgi; ð2Þ where a; b¼1;…; N denote field space and A; B¼ 1;…;4correspond to Dirac spinor indices. Fermion flavor indices are omitted and the operator nature of the quantum fields is implied. We consider systems with spatial isotropy and homogeneity such that the spectral functions depend on times and relative spatial coordinates, i.e.,ρðt; t0;jx−yjÞor in momentum spaceρðt; t0;jpjÞ, while the field expectation value only depends on time, i.e., hσðtÞi. Due to the remaining OðN−1Þ symmetry of the chirally broken model, the bosonic spectral function can be written as ρϕab¼diagðρσ;ρπ;ρπ;ρπÞ where the components ρi with i¼σ, π describe the respective mesons. The fermionic spectral function can be decomposed into Lorentz compo- nents according to
ρψ ¼ρSþiγ5ρPþγμρμVþγμγ5ρμAþ1
2σμνρμνT ; ð3Þ withσμν¼2i½γμ;γνandγ5¼iγ0γ1γ2γ3. The corresponding Lorentz componentsare given by
ρS¼1 4Tr½ρψ; ρP ¼1
4Tr½−iγ5ρψ; ρμV ¼1
4Tr½γμρψ; ρμA¼1
4Tr½γ5γμρψ;ρμνT ¼1
4Tr½σμνρψ; ð4Þ where the trace acts in Dirac space. In spatially homo- geneous and isotropic systems with parity and CPinvari- ance, the only nonvanishing components are the scalar, vector, and 0i-tensor components. Rotational invariance allows us to write
ρSðx0; y0;pÞ ¼ρSðx0; y0;jpjÞ;
ρ0Vðx0; y0;pÞ ¼ρ0ðx0; y0;jpjÞ;
ρiVðx0; y0;pÞ ¼ pi
jpjρVðx0; y0;jpjÞ;
ρ0iTðx0; y0;pÞ ¼ pi
jpjρTðx0; y0;jpjÞ; ð5Þ where we refer to the two-point functionsρS,ρ0,ρV, andρT
on the right-hand sides as the scalar, vector, vector-zero, and tensor components. The relevant contributions to the quark spectral function are the scalar, vector-zero, and vector components, where the vector-zero component represents the quark excitations of the system [25,34].
For chiral symmetric theories withmψ ¼0, the scalar and tensor components vanish. The spectral functions also encode the equal-time commutation and anticommutation relations of the quantum theory, implying that
i∂tρϕðt; t0;jpjÞjt¼t0¼1; ρ0ðt; t;jpjÞ ¼i; ð6Þ while all other fermion components vanish at equal time.
In addition to the spectral functions, we may consider the so-calledstatistical functions. These are the anticommuta- tor and commutator expectation values,
Fϕðx; yÞ ¼1
2hfφðxÞ;φðyÞgi−ϕðxÞϕðyÞ;
Fψðx; yÞ ¼1
2h½ψðxÞ;ψ¯ðyÞi; ð7Þ
where field space, Dirac, and flavor indices are suppressed.
The statistical functions carry information about the par- ticle density of the system, i.e., the occupation of the available modes in the system. Together, the spectral and statistical functions fully describe the time-ordered con- nected two-point correlation function, commonly denoted asGðx; yÞ ¼ hTφðxÞφðyÞi−hφðxÞihφðyÞifor the bosonic andΔðx; yÞ ¼ hTψðxÞψðyÞi¯ for the fermionic sector. Note that in nonequilibrium settings, the time-ordering occurs along the closed time path also known as Schwinger- Keldysh contour.
A. 2PI effective action real-time formalism at NLO One can derive closed and nonsecular evolution equa- tions for the one- and two-point functions of the quark- meson model out of equilibrium. These equations follow from the 2PI effective action Γ½ϕ; G;Δ, the quantum counterpart of the classical action S½ψ¯;ψ;σ;π, via a variational principle (see, e.g., [35]). The 2PI effective action of the quark-meson model can be written as
The relevant evolution equations for the one- and two- point functions have the form (explicit expressions can be found in the Appendix),
½□xþM2ðxÞϕðxÞ ¼ Z x0
0 dzΣϕðx; zÞϕðzÞ þJϕðxÞ;
½□xþM2ϕðxÞρϕðx; yÞ ¼ Z x0
y0
dzΣϕρðx; zÞρϕðz; yÞ;
½i=∂xþMψðxÞρψðx; yÞ ¼ Z x0
y0
dzΣψρðx; zÞρψðz; yÞ; ð9Þ
with shorthand notation Rt2
t1 dz≡Rt2
t1 dz0R
d3z and the dependence of the self-energies Σi onϕ; G;Δ is implied.
Similar expressions hold for the statistical functions. On the left-hand side, the Klein-Gordon or Dirac operators act on the corresponding expectation value. Thereby, effective masses take into account local quantum corrections. On the right, the effects of quantum fluctuations appear in so- called memory integrals that encode the generally non- Markovian effects of fluctuations in the past. The source term Jϕ in the field equation arises in the chirally broken case and describes the fermion backreaction on the field. It pushes the field to nonzero field expectation values even in the case where ϕðtÞ ¼∂tϕðtÞ ¼0at initial time.
In order to carry out explicit computations, the self- energiesΣi need to be approximated. Here we deploy an
the given approximation scheme are provided. To study the time evolution of the system, we iteratively solve the equations of motion without further approximations.
B. Initial conditions
The derivation of the nonequilibrium 2PI effective action and the equations of motion following from it rely on the assumption of a Gaussian initial state. This corresponds to a system initially exhibiting the characteristics of a non- interacting theory. However, higher order correlation func- tions build up during the subsequent time evolution. While this appears at first sight to correspond to a very limited choice of initial conditions, it still allows for a wide variety of different configurations through which we can determine for instance the energy densityεinitat the beginning of our computation. In particular, the Gaussian initial state rep- resents a genuine nonequilibrium state in the fully inter- acting nonequilibrium system, in which the time evolution takes place.
We allow for spontaneous symmetry breaking by using a negative mesonic bare mass squaredm2<0in the classical potential of the system. Since the initial state is determined by a free theory withm2¼m2init >0, the sign flip of m2 leads to a quench of the classical potential from positive to negative curvature in the first time step. At initial time, the classical potential is minimal at vanishing field expectation value while the minimum at t >0 becomes nonzero by takingm2<0.
FIG. 1. 2PI diagrams at NLO in1=Nandg. Full lines represent boson propagators, crossed circles macroscopic field insertions, and dashed lines fermion propagators. The first two-loop diagram in the first row corresponds to the leading-order contribution in1=N. The last diagram in the second row shows the fermion boson loop. The other diagrams in the first and second rows depict the infinite series of NLO diagrams in1=N.
A Gaussian initial state can be fully specified in terms of the one- and two-point functions. Since the field evolution equation involves second order time derivatives, one has to specify both the sigma field value and its initial time derivative. We choose the latter to vanish and refer to the initial field expectation value asσ0,
σðt¼0Þ ¼σ0; ∂tσðtÞjt¼0¼0; ð10Þ where σðtÞ now denotes the expectation value of the sigma field. As pointed out above, due to the presence of a finite bare quark massmψ, the field can move away fromσ0¼0due to the backreaction with the fluctuations of the theory.
We specify the initial conditions for the two-point functions in terms of the spectral and statistical compo- nents. The initial conditions for the bosonic (fermionic) spectral functions are fully determined by the equal-time commutation and (anti)commutation relations (6). For the remaining statistical functions, we employ free-field expressions with a given initial particle number. The bosonic statistical function then reads
Fiðt; t0;jpjÞ ¼niðt;jpjÞ þ12
ωiðt;jpjÞ cos½ωiðt;jpjÞðt−t0Þ; ð11Þ with i¼σ, π and where at initial time t¼t0¼0 the dispersion is set toωið0;jpjÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
jpj2þm2init
p with initial
mass squared m2init>0and the particle distribution given by nið0;jpjÞ ¼0. For the fermions, the free statistical function can be written as
Fψðt; t;jpjÞ ¼−γipiþmψ ωψðt;jpjÞ
1
2−nψðt;jpjÞ
; ð12Þ
where we choose the initial dispersion to be ωψð0;jpjÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
jpj2þm2ψ q
and the initial particle distribution to be constant, i.e., nψð0;jpjÞ ¼n0.
The energy contained in the initial state via εinit deter- mines the temperature at which the system thermalizes.
By preparing different initial conditions, we can study the thermalization process toward different temperatures and hence phases of the model as sketched in Fig. 2.
C. Numerical implementation
As is customary in the context of the 2PI effective action, we discretize the system on the level of the equations of motion (9). The explicit form of the equations allows us to deploy a leap-frog scheme, where in particular the fermionic two-point functions are dis- cretized in a temporally staggered fashion. The two- point functions, as the name suggests, carry an explicit dependence on two temporal coordinates. Since the
memory integrals contain the full time history, the required memory grows quadratically with the number of time steps. In order to keep the computation manage- able, we reduce the memory burden by exploiting isotropy and homogeneity, which reduces the effective spatial dimensions to one. A modified Fourier transform based on Hankel functions allows us to evaluate the self- energy contributions in coordinate space and to simplify the convolutions in the memory integrals in momentum space. For this project, we extended the code used in Ref. [2]to include the additional nonvanishing fermionic two-point functions present in our setup (the source code for this project is publicly accessible via the Zenodo repository under [37]).
In the spirit of effective field theories, we choose a UV cutoff at a high enough momentum scale. Below this scale, we consider quantum and statistical fluctuation within the 2PI framework. The ultraviolet parameters of our effective field theory are cutoff dependent and chosen such that physical observables, i.e., mass ratios and the pion decay constant, are reproduced.
The numerical time evolution is computed using a spatial grid withNx ¼200lattice points and a lattice spacing of ax ¼0.2. The time step size is chosen to beat¼0.05ax guaranteeing energy conservation at the level of a few percent for the times analyzed. In the following, all dimensionful quantities will be given in units of the pseudocritical temperature Tpc, which has the value Tpc¼1.3a−1x determined according to the procedure described in Sec. IV B; see Fig. 19. Subsequently, we usemto denote the dimensionless ratiom=Tpcand likewise for all other dimensionful quantities.
Interactions between the macroscopic field, the bosonic and the fermionic propagators lead to an exchange of energy between the different sectors. To observe an efficient energy exchange and equilibration process at computationally accessible times, it is necessary to study FIG. 2. Sketch of the setup deployed in this study. We consider the real-time evolution from nonequilibrium initial states char- acterized by an energy density sourced either through a finiteσ field expectation value (blue circle) or a nonzero occupancy of fermionic modes (orange triangle). Depending on the initial energy contained in the system, one of three discernible final states, the chiral broken phase, the crossover regime, or the (almost) symmetric phase is approached.
computations of the quark-meson model with functional methods and a physical ultraviolet cutoff ΛUV≈1GeV, (see, e.g.,[9,39]). In these computations, it can be shown that the self-interaction is of subleading relevance for the fluctuation dynamics, despite the large size of the classical coupling λ. In the present 2PI framework, the quantum interactions are obtained through an NLO resummation and for large occupancies or large classical coupling they can be shown to be small.
The functional equilibrium studies [9,39], as well as a comparison of the quark-meson model to QCD (see, e.g.,[10]) reveal that a one-to-one correspondence of the low-energy limits of both theories in quantitative approx- imations to the full dynamics in the quark-meson model either requires a far smaller UV cutoff for the latter or a systematic improvement of the model toward QCD-assisted low-energy effective theories[8,9]. In the present work, we restrict ourselves to studying the qualitative properties of the nonequilibrium dynamics as a first step.
When identifying the sigma field expectation as pion decay constant, we can reproducefπ< mπ< mq< mσ at low temperatures. At the lowest temperatures considered in this work, we findfπ=mπ≃0.65, which is very close to the phenomenologically known value of approximately 0.69, the meson mass ratio mσ=mπ≃1.75, smaller than the vacuum value of around 2.9 but expected to increase when going to lower temperatures, andmq=mπ¼1.45, being on the order of magnitude with zero-temperature value of 2.6.
Hence, we expect our findings to qualitatively reproduce the QCD dynamics. Note, however, that the meson mass ratio mσ=mπ<2 leads to another order of the thresholds for scattering processes, and hence respective difference in the spectral functions.
For the bosonic sector, we use vacuum initial condi- tions, i.e., nϕðt¼0;jpjÞ ¼0. The initial mass is fixed at m2init ¼0.0047. The fermion initial distribution is chosen to be constant nψðt¼0;jpjÞ ¼n0. We study simulations with fluctuation dominated initial conditions where the fermion number n0 is varied between 0 and 1 while the initial field value is σ0¼0. Furthermore, the field domi- nated initial conditions with a nonvanishing field value of σðt¼0Þ ¼σ0 between 0 and 2.0 with vanishing fermion numbern0¼0are investigated. Unless otherwise
decay widths, providing insight into the modification of the system due to the presence of a (non)equilibrium medium.
Our numerical simulations find clear indications for qua- siparticles in both the IR and UV, revealing the presence of additional light propagating fermion modes for temper- atures above the pseudocritical temperature.
It is convenient to analyze the spectral functions in the Wigner representation where the Fourier transformed spectral function can be interpreted as the density of states such that its structure provides information about the quasiparticle states of the system. Therefore, the temporal dependence of the unequal-time two-point correlation functions on the two timest and t0 is rephrased in terms of Wigner coordinates: the central timeτ¼ ðtþt0Þ=2and the relative timeΔt¼t−t0. The dynamics inΔtdescribes microscopic properties of the system while the evolution in τdescribes macroscopic properties governed by nonequili- brium characteristics of the system. In order to study the frequency spectrum of the spectral functions, we then apply a Wigner transformation to the propagators. This corre- sponds to a finite range Fourier transformation of the propagators with respect to the relative timeΔt, which is constrained by 2τ in initial value problems where t; t0≥0. As a result, we obtain the frequency space spectral function
ρðτ;ω;jpjÞ ¼ Z 2τ
−2τdΔteiωΔtρðτ;Δt;jpjÞ ð13Þ with analogous expressions for all statistical functions.
For a real and antisymmetric spectral function (as in the bosonic case and for the fermionic scalar, vector and tensor components) as well as for an imaginary and symmetric spectral function (as for the fermionic vec- tor-zero component), the Wigner transformρðτ;ω;jpjÞ is imaginary. Due to symmetry, it is sufficient to present the Wigner transformed spectral functions for positive frequencies ω. Since the frequency space spectral func- tions are imaginary in our definition, we always plot−iρ in the subsequent sections, thereby omitting the−iin the plot labels to ease notation.
The commutation and anticommutation relations(6)can be rephrased in frequency space,
Z dω
2πωρϕðτ;ω;jpjÞ ¼i;
Z dω
2πρ0ðτ;ω;jpjÞ ¼i; ð14Þ where they are referred to as sum rules. In our numerical computations, the bosonic and fermionic sum rules are satisfied at the level ofOð10−2ÞandOð10−6Þ, respectively.
A. Establishing thermal equilibrium at late times Before embarking on a detailed study of the dynamical approach to thermal equilibrium, we first ascertain that our simulations of the quark-meson model exhibit thermal- ization at late times. We do so by observing the dynamic emergence of the fluctuation-dissipation theorem. One needs to keep in mind that as discussed in[1], the idealized thermal equilibrium state cannot be reached in principle due to the time reversibility of the evolution equations.
The simulation approaches the state more and more closely over time and at some point becomes indistinguishable from it for a given resolution. Hence, we expect the computation to approach a steady state.
The fluctuation-dissipation theorem is reflected in a particular property of the spectral and statistical functions in thermal equilibrium: they are not independent of each other. In four-dimensional Fourier space, it reads
Fϕeqðω;pÞ ¼−i 1
2þnBEðωÞ
ρϕeqðω;pÞ;
Fψeqðω;pÞ ¼−i 1
2−nFDðωÞ
ρψeqðω;pÞ; ð15Þ
with nBEðωÞ ¼ ðeβω−1Þ−1 being the Bose-Einstein and nFDðωÞ ¼ ðeβωþ1Þ−1 the Fermi-Dirac distribution. In (15), the frequency ω is the Fourier conjugate to the relative time Δt¼t−t0 as the time dependence of Feq and ρeq can be fully described in terms of Δt due to the time-translation invariance of thermal equilibrium.
Out of equilibrium, the independence of F and ρ manifests itself in the fact that the ratio F=ρ in general carries a momentum dependence. The equilibrium relation (15)on the other hand allows us to define the generalized particle distribution function[35]
niðτ;ω;jpjÞ ¼iFiðτ;ω;jpjÞ ρiðτ;ω;jpjÞ1
2; ð16Þ with a negative (positive) sign for bosonic (fermionic) components and i¼σ;π; V. This kind of distribution function has been studied in the context of nonthermal fixed points in relativistic as well as nonrelativistic scalar field theories[40]. Considering(16)the approach of thermal equilibrium in a general nonequilibrium, time evolution setup is characterized byniðτ;ω;jpjÞ→nBE=FDðωÞ.
In Fig. 3, we show the time evolution of the particle distribution defined in(16)for low and high momenta (left and right columns). One can see that at late times (red curves) the same shape is approached for small and large momenta, whereas at early times the distribution functions differ from each other. This loss of momentum dependence is required for the thermalization process and reflects the emergence of the fluctuation-dissipation relation in the equilibrium state. From the late-time distributions shown in Fig. 3, one can already guess that thermal distribution functions are reached.
We also observe that the evolution of the effective particle number is different for fermions and bosons.
The bosonic distribution functionsnσ andnπ show strong oscillations along frequencies at low momenta whereas oscillations at high momenta are weak. Since the particle distributions are computed by taking the ratio of the statistical and spectral functions, ni plotted against ω essentially describes how similar the peaks shapes of F andρare. In the high-momentum range, we find that the quasiparticle peaks of the bosonic statistical and spectral functions resemble one another from early times on, while in the low-momentum range more time is required for the peak shapes to become aligned. In contrast, the quarks show an opposite behavior. Their distributions have much stronger frequency oscillations for large momenta than for small momenta, i.e., it takes longer for the high-momentum modes to approach a thermal distribution.
Putting the pieces together, we can see that a redistrib- ution of the occupancies in fermionic and bosonic degrees of freedom occurs during the nonequilibrium time evolu- tion. While the time scales to converge to thermal distri- bution functions depend on the particle species and the momentum modes, we find that the distribution functions all become stationary for timesτ≳100, reflecting the time- translation invariant property of thermal equilibrium.
Although Fig. 3 already indicates the approach of thermal distribution functions, we still need to prove whether our final state actually fulfils the fluctuation- dissipation theorem. For a quantitative analysis, we com- pute thegeneralized Boltzmann exponents,
Aiðτ;ω;jpjÞ ¼ln½n−1i ðτ;ω;jpjÞ 1; ð17Þ with positive (negative) sign for bosonic (fermionic) components and i¼σ;π; V. In thermal equilibrium, the fluctuation-dissipation theorem (15) requires these expo- nents to sufficeAiðτ;ω;jpjÞ ¼βω, implying in particular that they become independent of momentumjpjand timeτ, where the latter is fulfilled by our late-time states.
A linear fit of our simulation data for the generalized Boltzmann exponents to βω yields the thermalization temperature Tp¼β−1p , which can in general be τ depen- dent. An example for such a fit is presented on the left side of Fig.4. The plot shows that the Boltzmann exponent of all
three components i¼σ,π,V nicely fits to the same line with slopeβ. We compute the temperature averaged over all momenta to obtain Ti for each component. The system temperature denoted byT is taken to be the mean over all three components.
For every simulation, we compute the temperatures at each momentumjpjand study the momentum dependence of the obtained temperature Tp. As pointed out in [28], thermodynamic relations can become valid before real thermal equilibrium is attained, a phenomenon known as prethermalization. Thermal equilibrium is characterized by Tp being equal to some equilibrium temperature for all modesjpj. On the right side of Fig.4, the deviations from the mean thermalization temperatureT are plotted. As can be seen, the deviations are very small. Hence, the Boltzmann exponents at late times τ become momentum independent and the late-time states are thermal in the sense
that they fulfill the fluctuation-dissipation theorem. The thermalization temperatures for all simulations in this work have been determined at time τ¼130. For the example shown in Fig. 4, it was checked that the thermalization temperatures found in the time range betweenτ¼100and τ¼160 are constant at the level of Oð10−3Þ. We have checked for all simulations in this work that the temperature has reached a stationary value at timeτ¼130.
Having clarified the successful approach to quantum thermal equilibrium in our system, we are now able to study the differences during the out-of-equilibrium evolution leading to the thermal states in detail.
B. Nonequilibrium time evolution of the spectral and statistical functions
In this section, we study the dynamics of the thermal- ization process, starting from fluctuation or field dominated FIG. 3. We show the time evolution of the effective particle number defined in(16)for bosonic and fermionic components (rows) and two different momenta (left and right columns). At late times (red curve), the effective particle number becomes time and momentum independent and approaches the shape of a Bose-Einstein and Fermi-Dirac distribution, respectively. The shown data are interpolated using a cubic spline. Dimensionful quantities are given in units of the pseudocritical temperatureTpc (cf. Fig. 19).
initial conditions. We investigate the time evolution of the spectral and statistical functions and consider derived quantities such as particle masses and widths. While the initial conditions strongly influence the nonequilibrium dynamics taking place, the final states are universal and characterized by the initial energy densityεinit that trans- lates into a unique temperature.
The time evolution leads to the emergence of quasipar- ticle peaks in the spectral functions of both quark and mesons. The value of the particle mass and its decay width are a consequence of the interactions taking place among the microscopic degrees of freedom. While the initial states correspond to free particles, which would have a spectrum given by aδdistribution located at the mass parameters of the classical action, the scattering effects included in the nonequilibrium evolution lead to peaks with finite widths in the spectrum.
In Fig. 5, we present a representative set of fermionic spectral functions from the vector-zero channel, which describes the quark excitation spectrum[34,41]. The three columns correspond to three different field dominated initial conditions of increasing initial energy density, as sketched by the blue dots in Fig.2. The top row shows the Wigner space spectral function at the lowest available momentum (IR), the bottom row at the highest momentum (UV). We can identify several characteristic properties of these spectral functions from a simple inspection by eye.
In the UV, a single quasiparticle structure is present at all times and at all energy densities. With increasing energy density in the initial state, corresponding to an increasing final temperature, the position of the peak and its width increase. This is consistent with the expectation that a fermion in an energetic medium will be imbued with an in- medium mass (to lowest order in perturbation theory it would be proportional to the temperature). Higher energy densities go hand in hand with an increased chance of scattering between the fermion and the other medium
constituents, which also leads to a larger in-medium width.
In the UV, no qualitative difference exists between the broken, crossover, or symmetric phase behavior.
On the other hand, in the IR, a clear distinction between the crossover region and all other energy density regimes is visible. While we also find a single quasiparticle structure at low and high initial energy densities, in the crossover region at early times no well-defined peaks are present at all. Instead, as times passes, two structures emerge. One dominant peak is located where one would expect the usual quasiparticle excitation to reside, another peak sits close to the frequency origin, denoting a significantly lighter addi- tional propagating mode.
In general, we find that also for the other fermionic and bosonic spectral functions the approach of the equilibrium state depends on the initial conditions. In the presence of a nonzero initial field valueσ0, the spectral functions evolve differently than in the case whereσ0¼0but the fermion occupation is finite, i.e., when the initial state contains more energy in terms of fermion occupations. As pointed out in Fig.5, the most interesting dynamical features can be seen in the low-momentum area, which we therefore focus on during the following analysis.
1. High-energy densities
Here, we study the quark-meson model at high enough initial energy densities such that the late-time evolution thermalizes in the high-temperature phase, where chiral symmetry is restored. For our analysis, we compare two simulations starting from different initial conditions char- acterized by almost indistinguishable energy densities.
One is dominated by the fieldσ0¼1.36andn0¼0, while the other is dominated by fermion fluctuationsσ0¼0and n0¼0.8. The final states feature similar thermalization temperatures of T¼3.15 and T¼3.18, respectively.
However, since the initial states are very different from FIG. 4. Left: the generalized Boltzmann exponents defined in(17)shown as a function of frequencyωat a given momentumjpjfor bosonic and fermionic components. For better visibility, only every 39th data point is shown. Using a linear fit, one can determine the slopeβand hence the temperatureTfor each component. The temperatureTindicated in the plot is averaged over all momenta and the three components. Right: the relative deviation from the thermalization temperatureΔ¼ ðTi−TÞ=Tshown for all three components as a function of momentum. The results for the bosonic and fermionic sectors agree very well. In both plots, dimensionful quantities are given in units of the pseudocritical temperatureTpc (cf. Fig.19).
each other, the evolution toward thermal equilibrium takes significantly different paths.
For such high initial energy densities, the differences in the time evolution are most apparent in the bosonic sector.
This can be studied by looking at the bosonic spectral and statistical functions. Numerical results are shown in Fig. 6, where only the pion spectral and statistical functions are presented since the behavior of the sigma meson is analogous. The final states of both simulations (red curve) are characterized by the same peak shapes for both spectral and statistical functions. However, the functions at intermediate times exhibit a completely different behavior.
For field dominated initial conditions (left column in Fig. 6), the peak position of the spectral function moves toward smaller frequencies with time, which means that the mass of the quasiparticle state decreases during the time evolution. In addition, the nonzero initial field leads to large amplitudes in the pion statistical function at early times (lower left plot in Fig.6) which corresponds to relatively high occupancies in the bosonic sector compared to the final thermal distribution. These occupancies have to redistribute to other bosonic momentum modes jpj and the fermionic sector to let the system equilibrate.
This behavior can be readily understood from the micro- scopic evolution equations of the system. The finite-valued
initial field drives the fluctuations in the bosonic sector because it contributes to the bosonic self-energy at initial timet¼0. Since the nonequilibrium time evolution takes into account the full time history sincet¼0, these initial fluctuations not only play a role at initial time but also at intermediate times. Only at late times, the system loses memory about the details of the initial state. Since the macroscopic field only couples to the bosons directly but not to the fermions, the energy provided by the initial field is first turned into bosonic fluctuations before being transferred to fermionic modes. As a consequence, the thermalization of an initial state with nonzero initial field value shows rich dynamics in the bosonic spectra.
In contrast, for fluctuation dominated initial conditions (right column in Fig.6), one observes a continuous increase of the amplitudes of both spectral and statistical functions until the maximum is reached in the thermal state. If the initial energy density is provided via fermionic fluctuations, the thermal final state is found to be realized already at intermediate times.
The spectral functions can be used to deduce the dispersion relation and lifetimes of the corresponding quasiparticle species. Following [42], we assume for the moment that the spectral function decays exponentially and can be approximated as ρðt; t0;jpjÞ ¼e−γpjt−t0jω−1p
sin½ωpðt−t0Þ with a dispersion ωp and a damping rate FIG. 5. A representative selection of spectral functions from the fermion vector-zero channel in the infrared (top row) and the UV (bottom row) in three different regimes labeled by the temperatures of their final state. Each panel contains four curves indicating different snapshots along the thermalization trajectory. All three simulations employ field dominated initial conditions, i.e.,σ0>0and n0¼0. Dimensionful quantities are given in units of the pseudocritical temperatureTpc (cf. Fig. 19).
γp, which are both allowed to be τ dependent. The corresponding Wigner transform is given byρðτ;ω;jpjÞ ¼ ρBWðτ;ω;jpjÞ þδρðτ;ω;jpjÞ whereρBWdenotes the rela- tivistic Breit-Wigner function
ρBWðτ;ω;jpjÞ ¼ 2ωΓðτ;jpjÞ
½ω2−ω2ðτ;jpjÞ2þω2Γ2ðτ;jpjÞ; ð18Þ which describes a peak with width Γðτ;jpjÞ ¼2γpðτÞ at position ω¼ωðτ;jpjÞ. The term δρ∼expð−2τγpÞ describes boundary effects due to the finite integration range in (13). Sinceδρ decreases exponentially with τγp, this term is negligible for sufficiently large damping ratios and/or sufficiently late times[42]. Otherwise, the frequency space spectral function suffers under severe noise coming from boundary effects. For all times shown in this work, we find that boundary effects are irrelevant.
We observe that peak shapes of the bosonic spectral functions can be well approximated by the Breit-Wigner function (18). At some given time τ, performing Breit- Wigner fits of the spectral function at all momenta jpj yields the dispersion relation ωðτ;jpjÞ and the momen- tum-dependent widthΓðτ;jpjÞ. For initial states with high- energy densities, such as considered in this section, the spectral and statistical functions exhibit quasiparticle peak structures already at early times (see Fig. 6).
Consequently, it is possible to fit a Breit-Wigner function
to the spectral functions at any stage such that the time evolution of the dispersion relation ωiðτ;jpjÞ and momentum-dependent width Γiðτ;jpjÞ for i¼σ, π can be mapped out.
In the left plot of Fig. 7, we show the dispersion relation of the pion at different times τ encoded in the color scheme. A fit ofωðτ;jpjÞto the relativistic dispersion relation Z ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
jpj2þm2
p at various times τ yields the quasiparticle masses mðτÞ, which are shown in the inset.
In the following, the stationary late-time value is denoted asm. We note that the mass corresponds to the dispersion relation in the limit of vanishing momentum, i.e., m¼ωðτ;jpj→0Þ. The right plot of Fig. 7 displays the momentum-dependent width of the pion extracted from the Breit-Wigner fits. We find a plateau in the IR and a maximum in the UV. In analogy to the dispersion, where the quasiparticle mass describes the zero-momentum limit, we can extract the asymptotic value of the width in the limit of vanishing momentum,Γ¼Γðτ;jpj→0Þ. Since Γcor- responds to the width of the spectral function that is peaked at the quasiparticle mass, it can be viewed as the width of the quasiparticle. As the right plot in Fig.7indicates,Γis increasing with time.
We can now work out the differences observed in Fig.6 in a quantitative fashion. There is an apparent difference in the approach of the late-time values of the massmπand the widthΓπwhen comparing the time evolution starting from FIG. 6. Time evolution of the pion spectral and statistical functions shown for two different initial conditions at the smallest available momentum jpj ¼0.012. The left column shows a simulation deploying field dominated initial conditions withσ0¼1.36, the right column fluctuation dominated initial conditions withn0¼0.8. Both simulations lead to thermal states at temperatures where chiral symmetry is restored. Dimensionful quantities are given in units of the pseudocritical temperatureTpc (cf. Fig.19).
the two different initial conditions. The results are shown in Fig.8, where again only the pion data are shown because the sigma meson behaves accordingly.
For field dominated initial conditions, the effective mass of the pion meson decreases during the time evolution, whereas for fluctuation dominated initial conditions it grows, albeit only slightly. This is in accordance to the previous observation of the shifting peak position for field dominated initial conditions. It is important to note that the mass of the quasiparticles is not contained in the initial state, sinceminit is much smaller than the particle masses of the thermal state, but generated dynamically during the time evolution. The quasiparticle masses build up from the fluctuations con- tained in the self-energies. Since the nonzero initial field value leads to large bosonic self-energy contributions in the beginning of the time evolution, at early times the masses are larger than in the case of vanishing initial field.
The time dependence of the spectral width shown in the right plot of Fig.8can be understood in terms of the sum
rule(14)according to which the bosonic spectral functions are normalized. Due to the additional factor of ω in the integrand, which arises from the time derivative on one of the fields in the boson commutation relation, a larger mass automatically implies smaller widths. Consequently, the behavior of mass and width in the time evolution must be converse to each other.
After discussing the dynamics of the meson spectral and statistical functions at high initial energy densities, we now turn to the quark sector. After decomposing the Dirac structure of fermionic two-point functions and imposing symmetries, we are dealing with four components for the quark spectral and statistical functions, the scalar, vector- zero, vector, and tensor components as introduced in(5). Of these four components, the vector-zero component contains information about which states can be occupied [34,41].
Since it is normalized to unity according to the sum rule (14), the vector-zero component quark spectral function can be interpreted as the density of states for the quarks.
FIG. 8. Time evolution of the pion mass and the pion width in the limit jpj→0. Results are shown for field dominated initial conditions withσ0¼1.36andn0¼0(blue dots) as well as fluctuation dominated initial conditions withσ0¼0andn0¼0.8(orange triangles). Dimensionful quantities are given in units of the pseudocritical temperatureTpc (cf. Fig.19).
FIG. 7. Time evolution of the dispersion relation and the momentum-dependent width of the pion. The inset shows the time evolution of pion mass obtained from fits of the dispersion relation toZ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
jpj2þm2π
p at various timesτ, whereZ¼1.07is obtained for all times analyzed. The data are shown for field dominated initial conditions withσ0¼1.24andn0¼0. Dimensionful quantities are given in units of the pseudocritical temperatureTpc (cf. Fig.19).
We note that in a chiral symmetric theory with vanishing fermion bare mass one findsρS¼ρP¼ρμνT ¼0since only components in (4) that anticommute withγ5 are allowed.
Here, we consider a setup where chiral symmetry restora- tion takes place. For initial conditions with high-energy densities and the corresponding final states in the high- temperature chiral symmetric regime, the quark dynamics can be studied in terms of the vector-zero and vector components.
As was shown in Fig.5, for high-energy densities there is not much dynamics taking place in the excitation spectrum of the quarks. More insight can be gained by looking at the vector component which is presented in Fig.9for the same field or fluctuation dominated initial conditions as dis- cussed before for the bosons. The interesting case is again the evolution starting from field dominated initial con- ditions. The corresponding vector spectral function (upper left plot) shows that the peak position moves toward smaller frequencies, just as in the bosonic case. It indicates that the energy of both meson and quark quasiparticles decreases during the time evolution. However, it is impor- tant to note that—in contrast to the mesons—the amplitude of the fermion statistical function increases during the time evolution. As discussed before, the nonzero initial field leads to strong fluctuations and hence occupancies in the bosonic sector. It takes time for these fluctuations to be transferred to the fermionic sector, which is why we observe that the fermion occupation grows slowly during the time evolution.
For the fluctuation dominated initial conditions, we again observe that the spectral and statistical functions approach their late-time behavior very quickly. We con- clude that the available states and their occupation quickly approach their thermal final state if energy is provided in terms of particles rather than the field in the initial state.
2. Intermediate energy densities
From Fig. 5, we can see that the most interesting dynamics is taking place for systems thermalizing in the crossover region. Thus, we aim to study the evolution of the vector-zero quark spectral function for two simulations thermalizing in the cross-over region.
Again we compare two simulations employing field or fluctuation dominated initial conditions, respectively, but in this case we are able to probe initial conditions that lead to the same late-time state. When comparing the late-time field expectation value σ¯, the mass ratio mσ=mπ, and the temperatureTof the final state of these two simulations, we find that the respective quantities differ by less than 0.5%.
Also, the shape of the spectral and statistical functions in frequency space is the same for bosonic as well as fermionic components. Quantitatively, we find that jρ1−ρ2j=maxðρ1Þis smaller thanOð10−2Þfor all frequen- ciesωand momentajpj, where the indices 1 and 2 denote the two simulations compared and maxðρÞ the maximal amplitude of ρ. Larger deviations are observed for the vector-zero component statistical function and for the FIG. 9. Time evolution of the vector component quark spectral and statistical functions shown for the same initial conditions as in Fig.6 at momentumjpj ¼0.016. Dimensionful quantities are given in units of the pseudocritical temperatureTpc (cf. Fig. 19).
tensor component spectral and statistical functions, where the amplitudes are of order Oð10−7Þ such that numerical inaccuracies come into play. In conclusion, we consider the late-time state of the two simulations to be the same thermal state, universal in the sense that the dependence on the initial conditions is lost. It is characterized solely by a temperature of T¼1.04, a mass ratio of mσ=mπ¼1.46, and a field expectation value of σ¯ ¼0.33. As we will see later, this corresponds to a state in the crossover region.
The regime of intermediate energy densities distin- guishes itself from high- and low-energy density initial conditions by showing a double-peak structure in the quark spectral functions. Our findings in a nonperturbative real- time setting corroborate previous observations of such double peak structures with perturbative computations or spectral reconstructions reported, e.g., in [34,43–49].
First, consider the vector-zero component describing the excitation spectrum of the quarks. In Fig.10, we show the time evolution of both spectral and statistical functions. As before, for fluctuation dominated initial conditions (right column) the system quickly approaches the shape of the late-time two-point functions. However, in the case of field dominated initial conditions, the double-peak structure of the spectral function only emerges at later times. At early times, the spectral function reveals a single broad structure.
We further point out that the statistical function F0 decays to zero during the time evolution, implying that the
fermion occupation is not contained in the vector-zero component but in other components. This agrees well with the effective quasiparticle number that has been employed previously[3,41],
nψðt;jpjÞ ¼1
2−jpjFVðt;t;jpjÞþMψðtÞFSðt;t;jpjÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
jpj2þM2ψðtÞ
q ; ð19Þ
with effective massMψðtÞ ¼mψ þhσðtÞ. This definition of an effective particle number only provides a good description of the quark content in the system if the occupations in the vector-zero and tensor component are negligible. In our computations, we find thatF0andFTare of the order Oð10−7Þ and hence irrelevant for the quark particle number.
In order to study the particle content, we take into account the vector component which is shown in Fig.11.
We can see that the double-peak structure observed in the vector-zero component is also visible in the vector com- ponent, in particular in both spectral and statistical func- tions. From this, we learn that the additional light degrees of freedom, provided in the low-frequency peak of the quark spectral density, is actually occupied in terms of the vector component quark statistical function. Hence, for states thermalizing in the crossover temperature regime, FIG. 10. Time evolution of the vector-zero component quark spectral and statistical functions shown for two different initial conditions at momentum jpj ¼0.012. The left column shows field dominated initial conditions with σ0¼0.98, the right column fluctuation dominated initial conditions withn0¼0.11. Both lead to the same late-time state withT¼1.04. Dimensionful quantities are given in units of the pseudocritical temperatureTpc (cf. Fig.19).
there is an additional light mode with finite occupation in the quark sector available to participate in the dynamics.
We further observe that for fixed momentum jpj the energy of the light mode increases with rising temperature.
At sufficiently high temperatures, this additional mode reaches energies comparable with the main quasiparticle mode such that the two peaks merge into the single peak persistent in the high-temperature regime.
We conclude this section with a comment on the dynamics found for initial states with low-energy densities.
In contrast to the cases of intermediate and high-energy densities, we find well-defined quasiparticle peaks for both quarks and mesons. The smaller energy density leads to lower thermalization temperatures and a stronger chiral symmetry breaking, reflected by a mass difference between the σ andπ mesons. After discussing the nonequilibrium time evolution of the spectral functions, we now turn to the equilibrium properties.
C. Late-time thermal limit
In this section, we discuss the spectral functions of quarks and mesons in the state of quantum thermal equilibrium according to the definition introduced in Sec. III A. The properties of spectral functions at dif- ferent temperatures reflect the crossover transition of the quark-meson model from the chiral broken to a chiral
symmetric phase. We find that the shapes of the final states are universal in the sense that they only depend on the temperature and not on the details of the initial state.
1. Mesons
Information about the different phases of the model can be obtained from the temperature dependence of the late- time thermal spectral functions of the mesons. We find that the shape of the bosonic spectral functions is described by a Breit-Wigner function for all considered temperatures.
Thereby, the width and the position of the Breit-Wigner peak only depend on the temperature but not on the initial conditions chosen.
As discussed in Sec. III B 1, the momentum-dependent width and the dispersion relation are obtained by applying Breit-Wigner fits to the spectral functions. Although the Breit-Wigner function(18) has two parameters, the width Γðτ;jpjÞand the peak position given byωðτ;jpjÞ, there is only one free parameter since the normalization condition given by the sum rule(14)must be satisfied.
In the right plot of Fig.7, we already saw that there is a characteristic momentum modejpj at which the momen- tum-dependent width becomes maximal. This corresponds to the momentum at which the decay is strongest and can be considered as the main decay mode, in the following denoted byQ. In the left plot of Fig.12, we show the main FIG. 11. Time evolution of the vector component quark spectral and statistical functions shown for two different initial conditions at momentum jpj ¼0.012. The left column shows field dominated initial conditions with σ0¼0.98, the right column fluctuation dominated initial conditions withn0¼0.11. Both lead to the same late-time state withT¼1.04. Dimensionful quantities are given in units of the pseudocritical temperatureTpc
decay modeQas a function of temperature for both meson species. At low temperatures, the strongest decays are found in the IR, whereas at high temperatures the strongest decays occur in the UV. There is an abrupt change at some critical temperature, above whichQ >0meaning that the momentum-dependent width has a maximum at a nonzero momentum, as shown by the upper line in the inset.
Comparing the momentum-dependent width at low T and highT, we can see that the transition from the chiral broken to the chiral symmetric phase is characterized by new decay modes in the UV. Thereby, the main decay mode is suddenly shifted from the IR to the UV.
Another prominent signature for the crossover transition is provided by the quasiparticle masses of the σ and π mesons. The two meson species are distinguishable in the chiral broken phase, where they have different masses, while they become identical in the chiral symmetric phase.
When plotting the meson masses as a function of temper- ature, as shown in the right plot of Fig.12, we can nicely visualize the restoration of chiral symmetry, manifest in the quasiparticle masses of σ andπ becoming identical (pink and cyan data points). We observe a softening of the masses at intermediate temperatures, i.e., the quasiparticle masses are minimal in the temperature region where the crossover phase transition occurs. Decreasing masses indicate grow- ing correlation lengths. In the limit of a second order phase transition, which is characterized by diverging correlation lengths, the masses would vanish at the transition point.
In the high-temperature range, masses grow with rising temperatures. This reflects that the quasiparticle masses can be considered asthermal massesin the sense that they
contain self-energy contributions and are generated by quantum fluctuations, which increase with temperature.
We further note that one could also study the temperature dependence of the width Γ¼Γðτ;jpj→0Þ instead of m¼ωðτ;jpj→0Þ. However, the information is equivalent due to the normalization of the spectral functions, as pointed out above. Consequently, the behavior of Γ is converse to the behavior of m and not presented here explicitly. The width Γ is small at low temperatures, strongly grows toward intermediate temperatures where it reaches a maximum value in the crossover temperature regime, and then decays slowly when going to higher temperatures.
2. Quarks
Let us now consider the thermal spectral functions for the quark sector. Several aspects of the different components invite for discussion. Let us begin with a recap of the findings shown in the vector-zero component of the quark spectral function. As presented in Fig. 5, the spectral density has different shapes at low, intermediate, and high temperatures. In particular, the intermediate temperature range of the crossover transition is characterized by a double-peak structure. The temperature dependence of the fermionic quasiparticle masses is depicted in Fig.12.
The mass of the low-frequency mode (plasmino branch, denoted by p) grows continuously with rising T until it merges with the main peak (denoted byq), forming the wide quasiparticle peak found for initial states with large energy densities. For related studies with perturbative computations or spectral reconstructions, see, e.g.,[34,43–49]. Note also FIG. 12. Left: temperature dependence of the characteristic decay momentumQshown for the σand πmesons. The inset shows examples for the momentum-dependent width at high and low temperatures.Qcorresponds to the momentum at which the width Γðτ;jpjÞis maximal. Right: temperature dependence of quasiparticle masses. Restoration of chiral symmetry is reflected in identical masses of theσandπmesons at high temperatures. The quarkqquasiparticle mass is obtained from the dominant peak of the vector- zero component quark spectral function. We also plot the“plasmino”branchpobtained from the quark spectral function. In both plots, gray lines show cubic spline interpolations of the data points. Dimensionful quantities are given in units of the pseudocritical temperatureTpc (cf. Fig. 19).
